Naturally reductive Killing \(f\)-manifolds. (English. Russian original) Zbl 0977.53025

Balashchenko, V. V.

Russ. Math. Surv. 54, No. 3, 623-625 (1999); translation from Usp. Mat. Nauk 54, No. 3, 151-152 (1999).

From the text: Amongst the most remarkable almost Hermitian structures \((g,J)\) on smooth manifolds \(M\) are the nearly Kähler structures. The defining condition is the requirement that \(\nabla_X(J)X=0\), where \(\nabla\) is the Levi-Civita connection of the (pseudo) Riemannian metric \(g\) and \(X\) is an arbitrary smooth vector field on \(M\). The Killing \(f\)-structures are a generalization of this concept to metric \(f\)-structures of classical type (those with \(f^3+f=0)\), and the defining relation here is \(\nabla_X (f)X=0\). In this article we analyse the possible existence of invariant Killing \(f\)-structures on naturally reductive homogeneous spaces. In particular, criteria are obtained for canonical \(f\)-structures on homogeneous \(\Phi\)-spaces of orders 4 and 5 to be Killing.

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C55	Global differential geometry of Hermitian and Kählerian manifolds
53C30	Differential geometry of homogeneous manifolds

Keywords:

Killing \(f\)-structures; metric \(f\)-structures; nearly Kähler structures; naturally reductive homogeneous spaces